Connection is module structure over connection algebra: Difference between revisions

Revision as of 00:18, 24 July 2009

Statement

Let $E$ be a vector bundle over a differential manifold $M$ . Then, a connection on $E$ is equivalent to giving $\Gamma (E)$ (the vector space of sections of $E$ ) the structure of a module over the connection algebra of $M$ . Equivalently, it gives ${\mathcal {E}}$ (the sheaf of sections of $E$ ) the structure of a module over the sheaf of connection algebras over $M$ .

Definitions used

Connection

Further information: Connection

Connection algebra

Further information: Connection algebra

Proof

From a connection to a module structure

The outline of the proof is as follows:

We first show that a connection gives an action of the first-order differentiable operators on the space of sections.
Next, we show that the Leibniz rule property of connections allows us to extend this to a well-defined action of the connection algebra.

Given: A manifold $M$ , a vector bundle $E$ over $M$ , a connection $\nabla$ on $E$ . $B$ is the algebra of smooth fiber-preserving maps from $\Gamma (E)$ to $\Gamma (E)$ . ${\mathcal {D}}^{1}(M)$ is the Lie algebra of first-order differential operators on $M$ and ${\mathcal {C}}(M)$ is the connection algebra on $M$ .

To prove: $\nabla$ gives rise to a homomorphism from ${\mathcal {C}}(M)$ to $B$ .

Proof: $\nabla$ gives rise to a map:

$f_{\nabla }:D^{1}(M)\to B$

as follows:

$f_{\nabla }(X+m(g))=s\mapsto \nabla _{X}(s)+(gs)$ .

First observe that the map sends $C^{\infty }(M)\subset {\mathcal {D}}^{1}(M)$ to $C^{\infty }(M)\subset B$ , and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function $f$ , goes to the operator of multiplication by the function $f$ .

We now prove some basic results about $f_{\nabla }$ :

$f_{\nabla }$ is $\mathbb {R}$ -bilinear: This is obvious.
For any element $X+m(g)$ in ${\mathcal {D}}^{1}(M)$ and any $h\in C^{\infty }(M)$ , we have $f_{\nabla }(m(g){\dot {(}}X+m(h))(s)=m(g)f_{\nabla }(X+m(h))(s)$ . This essentially follows from the fact that a connection is tensorial in the direction of differentiation:

$f_{\nabla }(m(g){\dot {(}}X+m(h)))(s)=f_{\nabla }(gX+m(gh))(s)=\nabla _{gX}(s)+(gh)(s)=g\nabla _{X}(s)+(gh)(s)=g(\nabla _{X}(s)+hs)$ .

For any element $X+m(g)$ in ${\mathcal {D}}^{1}(M)$ and any $h\in C^{\infty }(M)$ , we have $f_{\nabla }((X+m(h)){\dot {m}}(g))(s)=f_{\nabla }(X+m(h))(m(g)s)$ . This essentially follows from the Leibniz rule property.

$f_{\nabla }((X+m(h)){\dot {m}}(g))(s)=f_{\nabla }(m(Xg)+g\nabla _{X}+m(gh))(s)=(Xg)(s)+g\nabla _{X}(s)+(gh)s=\nabla _{X}(gs)+(gh)(s)$ .

References

Textbook references

Book:Globalcalculus^{More info}, Page 64

@@ Line 13: / Line 13: @@
 ==Proof==
-We start with a connection <math>\nabla</math> on <math>E</math> and show how <math>\nabla</math> naturally equips <math>E</math> with the structure of a module over <math>\mathcal{C}(M)</math>.
+===From a connection to a module structure===
-Let <math>D^1(M)</math> denote the [[Lie algebra of first-order differential operators]] of <math>M</math>. For now, we're thinking of <math>D^1(M)</math> as a <math>C^\infty(M)</math>-bimodule. Let <math>B</math> be the algebra of all smooth fiber-preserving linear maps from <math>\Gamma(E)</math> to <math>\Gamma(E)</math>.
+The outline of the proof is as follows:
-A connection gives a map:
+* We first show that a connection gives an action of the first-order differentiable operators on the space of sections.
+* Next, we show that the Leibniz rule property of connections allows us to extend this to a well-defined action of the connection algebra.
-<math>D^1(M) \to B</math>
+'''Given''': A manifold <math>M</math>, a vector bundle <math>E</math> over <math>M</math>, a connection <math>\nabla</math> on <math>E</math>. <math>B</math> is the algebra of smooth fiber-preserving maps from <math>\Gamma(E)</math> to <math>\Gamma(E)</math>. <math>\mathcal{D}^1(M)</math> is the Lie algebra of first-order differential operators on <math>M</math> and <math>\mathcal{C}(M)</math> is the connection algebra on <math>M</math>.
-as follows:
+'''To prove''': <math>\nabla</math> gives rise to a homomorphism from <math>\mathcal{C}(M)</math> to <math>B</math>.
-<math>X \mapsto (s \mapsto \nabla_X(s))</math>
+'''Proof''': <math>\nabla</math> gives rise to a map:
-First observe that the map sends <math>C^\infty(M) \subset D^1(M)</math> to <math>C^\infty(M) \subset B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>.
+<math>f_\nabla: D^1(M) \to B</math>
-We now argue that the map is in fact a <math>C^\infty(M)</math>-bimodule map. The fact that it is a left <math>C^\infty(M)</math>-module map follows from the fact that:
+as follows:
-<math>\nabla_{fX} = f \nabla_X</math>
+<math>f_\nabla(X+m(g)) = s \mapsto \nabla_X(s) + (gs)</math>.
-while the fact that it is a right <math>C^\infty(M)</math>-module map follows from the Leibniz rule.
+First observe that the map sends <math>C^\infty(M) \subset \mathcal{D}^1(M)</math> to <math>C^\infty(M) \subset B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>.
-The upshot is that the induced map:
+We now prove some basic results about <math>f_\nabla</math>:
-<math>D^1(M) \to B</math>
+* <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious.
+* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \dot (X + m(h))(s) = m(g)f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map|tensorial]] in the direction of differentiation:
-is a <math>C^\infty(M)</math>-bimodule map, and hence extends to a <math>\R</math>-linear homomorphism from the tensor algebra of <math>D^1(M)</math> over <math>C^\infty(M)</math> to <math>B</math>. Clearly, <math>m(1) - 1</math> acts trivially, so we obtain a homomorphism:
+<math>f_\nabla(m(g) \dot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs)</math>.
-<math>\mathcal{C}(M) \to B</math>
+* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have  <math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(X + m(h))(m(g)s)</math>. This essentially follows from the Leibniz rule property.
-giving <math>\Gamma(E)</math> the structure of a module over the algebra <math>\mathcal{C}(M)</math>.
+<math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(m(Xg) +g\nabla_X + m(gh))(s) = (Xg)(s) + g\nabla_X(s) + (gh)s = \nabla_X(gs) + (gh)(s)</math>.
 ==References==